The paper "On the constant in a transference inequality for the vector-valued Fourier transform" revisited

TL;DR

This paper provides a concise proof that the functions involved in Fourier transform inequalities reach their global minimum at the midpoint, refining the understanding of constants in transference inequalities for vector-valued Fourier transforms.

Contribution

It offers a short proof confirming the global minimum of specific functions at the midpoint, clarifying the constant in a key transference inequality for Fourier transforms.

Findings

The functions f_r attain their minimum at x=1/2.

The proof simplifies understanding of constants in Fourier type equivalence.

Clarifies the behavior of functions related to Fourier transform inequalities.

Abstract

The standard proof of the equivalence of Fourier type on \(\mathbb R^d\) and on the torus \(\mathbb T^d\) is usually stated in terms of an implicit constant which can be expressed in terms of the global minimiser of the functions \[f_r(x)=\sum_{m\in\mathbb{Z}}\left|\frac{\sin(\pi(x+m))}{\pi(x+m)}\right|^{2r},\qquad x\in [0,1], \ r\ge 1.\] The aim of this note is to provide a short proof of a result of the authors which states that each \(f_r\) takes a global minimum at the point \(x = \frac12\).